“Onsager–Machlup function”的意思、由来-开放百科全书

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Onsager–Machlup function

释义

Definition
Examples
Wiener process on the real line Diffusion processes with constant diffusion coefficient on Euclidean space
Generalizations
Applications
See also
References
Bibliography
External links

The Onsager–Machlup function is a function that summarizes the dynamics of a continuous stochastic process. It is used to define a probability density for a stochastic process, and it is similar to the Lagrangian of a dynamical system. It is named after Lars Onsager and S. Machlup who were the first to consider such probability densities.^[1]

The dynamics of a continuous stochastic process {{mvar|X}} from time {{math|t {{=}} 0}} to {{math|t {{=}} T}} in one dimension, satisfying a stochastic differential equation

where {{mvar|W}} is a Wiener process, can in approximation be described by the probability density function of its value {{math|x_i}} at a finite number of points in time {{math|t_i}}:

where

and {{math|Δt_i {{=}} t_i+1 − t_i > 0}}, {{math|t₁ {{=}} 0}} and {{math|t_n {{=}} T}}. A similar approximation is possible for processes in higher dimensions. The approximation is more accurate for smaller time step sizes {{math|Δt_i}}, but in the limit {{math|Δt_i → 0}} the probability density function becomes ill defined, one reason being that the product of terms

diverges to infinity. In order to nevertheless define a density for the continuous stochastic process {{mvar|X}}, ratios of probabilities of {{mvar|X}} lying within a small distance {{mvar|ε}} from smooth curves {{math|φ₁}} and {{math|φ₂}} are considered:^[2]

as {{math|ε → 0}}, where {{mvar|L}} is the Onsager–Machlup function.

Definition

Consider a {{mvar|d}}-dimensional Riemannian manifold {{mvar|M}} and a diffusion process {{math|X {{=}} {X_t : 0 ≤ t ≤ T} }} on {{mvar|M}} with infinitesimal generator {{math|{{sfrac|1|2}}Δ_M + b}}, where {{math|Δ_M}} is the Laplace–Beltrami operator and {{mvar|b}} is a vector field. For any two smooth curves {{math|φ₁, φ₂ : [0, T] → M}},

where {{mvar|ρ}} is the Riemannian distance, denote the first derivatives of {{math|φ₁, φ₂}}, and {{mvar|L}} is called the Onsager–Machlup function.

The Onsager–Machlup function is given by^[3]^[4]^[5]

where {{math|{{!!}} ⋅ {{!!}}_x}} is the Riemannian norm in the tangent space {{math|T_x(M)}} at {{mvar|x}}, {{math|div b(x)}} is the divergence of {{mvar|b}} at {{mvar|x}}, and {{math|R(x)}} is the scalar curvature at {{mvar|x}}.

Examples

The following examples give explicit expressions for the Onsager–Machlup function of a continuous stochastic processes.

Wiener process on the real line

The Onsager–Machlup function of a Wiener process on the real line {{math|R}} is given by^[6]

[Proof]

Let {{math|X {{=}} {X_t : 0 ≤ t ≤ T} }} be a Wiener process on {{math|R}} and let {{math|φ : [0, T] → R}} be a twice differentiable curve such that {{math|φ(0) {{=}} X₀}}. Define another process {{math|X^φ {{=}} {X_t^φ : 0 ≤ t ≤ T} }} by {{math|X_t^φ {{=}} X_t − φ(t)}} and a measure {{math|P^φ}} by

For every {{math|ε > 0}}, the probability that {{math|{{!}}X_t − φ(t){{!}} ≤ ε}} for every {{math|t ∈ [0, T]}} satisfies

By Girsanov's theorem, the distribution of {{math|X^φ}} under {{math|P^φ}} equals the distribution of {{mvar|X}} under {{mvar|P}}, hence the latter can be substituted by the former:

By Itō's lemma it holds that

where is the second derivative of {{mvar|φ}}, and so this term is of order {{mvar|ε}} on the event where {{math|{{!}}X_t{{!}} ≤ ε}} for every {{math|t ∈ [0, T]}} and will disappear in the limit {{math|ε → 0}}, hence

Diffusion processes with constant diffusion coefficient on Euclidean space

The Onsager–Machlup function in the one-dimensional case with constant diffusion coefficient {{mvar|σ}} is given by^[7]

In the {{mvar|d}}-dimensional case, with {{mvar|σ}} equal to the unit matrix, it is given by^[8]

where {{math|{{!!}} ⋅ {{!!}}}} is the Euclidean norm and

Generalizations

Generalizations have been obtained by weakening the differentiability condition on the curve {{mvar|φ}}.^[9] Rather than taking the maximum distance between the stochastic process and the curve over a time interval, other conditions have been considered such as distances based on completely convex norms^[10] and Hölder, Besov and Sobolev type norms.^[11]

Applications

The Onsager–Machlup function can be used for purposes of reweighting and sampling trajectories,^[12]

as well as for determining the most probable trajectory of a diffusion process.^[13]^[14]

References

1. ^Onsager, L. and Machlup, S. (1953)
2. ^Stratonovich, R. (1971)
3. ^Takahashi, Y. and Watanabe, S. (1980)
4. ^Fujita, T. and Kotani, S. (1982)
5. ^Wittich, Olaf
6. ^Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9
7. ^Dürr, D. and Bach, A. (1978)
8. ^Ikeda, N. and Watanabe, S. (1980), Chapter VI, Section 9
9. ^Zeitouni, O. (1989)
10. ^Shepp, L. and Zeitouni, O. (1993)
11. ^Capitaine, M. (1995)
12. ^Adib, A.B. (2008).
13. ^Adib, A.B. (2008).
14. ^Dürr, D. and Bach, A. (1978).

Bibliography

{{Cite journal | author = Adib, A.B. | year = 2008 | title = Stochastic actions for diffusive dynamics: Reweighting, sampling, and minimization | journal = J. Phys. Chem. B | volume = 112 | issue = 19 | pages = 5910–5916 | doi=10.1021/jp0751458| pmid = 17999482 | arxiv =0712.1255 }}
{{Cite journal | author = Capitaine, M. | year = 1995 | title = Onsager–Machlup functional for some smooth norms on Wiener space | journal = Probab. Theory Relat. Fields | volume = 102 | issue = 2 | pages = 189–201 | doi=10.1007/bf01213388}}
{{Cite journal |author1=Dürr, D. |author2=Bach, A. |lastauthoramp=yes | year = 1978 | title = The Onsager–Machlup function as Lagrangian for the most probable path of a diffusion process | journal = Commun. Math. Phys. | volume = 60 |issue=2 | pages = 153–170 | doi=10.1007/bf01609446|bibcode=1978CMaPh..60..153D}}
{{Cite journal |author1=Fujita, T. |author2=Kotani, S. |lastauthoramp=yes | year = 1982 | title = The Onsager–Machlup function for diffusion processes | journal = J. Math. Kyoto Univ. | volume = 22 | pages = 115–130|doi=10.1215/kjm/1250521863 }}
{{Cite book |author1=Ikeda, N. |author2=Watanabe, S. |lastauthoramp=yes | year = 1980 | title = Stochastic differential equations and diffusion processes | publisher = Kodansha-John Wiley}}
{{Cite journal |author1=Onsager, L. |author2=Machlup, S. |lastauthoramp=yes | year = 1953 | title = Fluctuations and Irreversible Processes | journal = Physical Review | volume = 91| number = 6 | pages = 1505–1512 | doi=10.1103/physrev.91.1505|bibcode=1953PhRv...91.1505O}}
{{Cite book |author1=Shepp, L. |author2=Zeitouni, O. |lastauthoramp=yes | year = 1993 | title = Exponential estimates for convex norms and some applications | journal = Progress in Probability | volume = 32 | pages = 203–215 | location = Berlin: Birkhauser-Verlag | doi=10.1007/978-3-0348-8555-3_11|isbn=978-3-0348-9677-1 |citeseerx=10.1.1.28.8641 }}
{{Cite journal | author = Stratonovich, R. | year = 1971 | title = On the probability functional of diffusion processes | journal = Select. Transl. In Math. Stat. Prob. | volume = 10 | pages = 273–286}}
{{Cite journal |author1=Takahashi, Y. |author2=Watanabe, S. |lastauthoramp=yes | year = 1980 | title = The probability functionals (Onsager–Machlup functions) of diffusion processes | journal = Lecture Notes in Mathematics | volume = 851| pages = 432–463 | publisher = Springer}}
{{Cite journal | author = Wittich, Olaf | title = The Onsager–Machlup Functional Revisited}}
{{Cite journal | author = Zeitouni, O. | year = 1989 | title = On the Onsager–Machlup functional of diffusion processes around non {{math|C²}} curves | journal = Annals of Probability | volume = 17 | number = 3 | pages = 1037–1054 | doi=10.1214/aop/1176991255}}

External links

Onsager–Machlup function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Onsager-Machlup_function&oldid=22857

3 : Functional analysis|Functions and mappings|Stochastic processes

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